Nonlinear Long-Wavelength Torsional Alfvén Waves

A&A 526, A80 (2011) Astronomy DOI: 10.1051/0004-6361/201016063 & c ESO 2010 Astrophysics

Nonlinear long-wavelength torsional Alfvén waves

S. Vasheghani Farahani1,V.M.Nakariakov1,2, T. Van Doorsselaere3, and E. Verwichte1

1 Centre for Fusion, Space and Astrophysics, Physics Department, University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected] 2 Central Astronomical Observatory of the Russian Academy of Sciences at Pulkovo, 196140 St Petersburg, Russia 3 Centrum voor Plasma-Astrofysica, Mathematics Department, Celestijnenlaan 200B bus 2400, 3001 Leuven, Belgium

Received 3 November 2010 / Accepted 23 November 2010 ABSTRACT

Aims. We investigate the nonlinear phenomena accompanying long-wavelength torsional waves in solar and stellar coronae. Methods. The second order thin flux-tube approximation is used to determine perturbations of a straight untwisted and non-rotating magnetic flux-tube, nonlinearly induced by long-wavelength axisymmetric magnetohydrodynamic waves of small, but finite amplitude. Results. Propagating torsional waves induce compressible perturbations oscillating with double the frequency of the torsional waves. In contrast with plane shear Alfvén waves, the amplitude of compressible perturbations is independent of the plasma-β and is proportional to the torsional wave amplitude squared. Standing torsional waves induce compressible perturbations of two kinds, that grow with the characteristic time inversely proportional to the sound speed, and that oscillate at double the frequency of the inducing torsional wave. The growing density perturbation saturates at the level, inversely proportional to the sound speed. Key words. magnetohydrodynamics (MHD) – waves – Sun: corona – Sun: activity – magnetic fields

1. Introduction The compressible flows induced by nonlinear Alfvén waves have been intensively studied in the context of solar wind accel- Alfvén waves are often considered as the primary candidate for eration (e.g. Ofman & Davila 1998). A modified Cohen-Kulsrud the acceleration of solar (e.g. Cranmer 2009) and stellar (e.g. equation describing spherical Alfvén waves in a stratified atmo- Charbonneau & MacGregor 1995; Suzuki 2007) winds and coro- sphere with a radial magnetic field was derived in Nakariakov nal heating (e.g. Ofman 2005). Alfvén waves are also considered et al. (2000b) in application to coronal holes. Analytical results in the context of the collimation and confinement of astrophys- are consistent with numerical MHD modelling performed in ical jets (Bisnovatyi-Kogan 2007), and in core-collapse super- Torkelsson & Boynton (1998). More advanced one-dimensional nova explosions (Suzuki et al. 2008). However, observational models for nonlinear Alfvén waves in solar and stellar atmo- ´ spheres in open-field regions include upflows, super-radial mag- evidence of Alfven waves in astrophysical plasmas still remains ff indirect, e.g. as a possible interpretation of non-thermal broad- netic field geometry, non-adiabatic e ects and various dissipa- ening of coronal emission lines (e.g. Banerjee et al. 2009)in tion mechanisms (Suzuki 2004; Suzuki & Inutsuka 2005; Suzuki coronal holes. 2007, 2008). Secularly growing compressible perturbations, induced by standing Alfvén waves, have been considered in the One of the key ingredients of the theoretical modelling of context of coronal loop hydrostatics by Litwin & Rosner (1998), Alfvén waves in solar and stellar atmospheres is the concept and in connection with magnetospheric field-line resonances by of the nonlinear cascade. It is needed to explain the transfer Tikhonchuk et al. (1995). Secular compressible effects associ- of wave energy from the low-frequency injection range (that is ated with travelling in opposite directions Alfvén waves were around 10 mHz in the solar atmosphere) to the high- found in Verwichte et al. (1999). All those studies were carried frequencydissipation range. In one-dimensional models, the out in terms of a shear Alfvén wave, one-dimensional model. nonlinear cascade is connected with nonlinear generation of One-dimensional models mentioned above are based upon higher harmonics, and hence steepening of the waves, caus- the assumption that the waves are plane. For long-period Alfvén ing the onset of nonlinear dissipation or of non-MHD dissipa- waves, with periods of the order of typical time scales of lower- tive processes. In a uniform plasma, this process is analytically atmospheric dynamics, this condition is not fulfilled. For exam- described by a weakly-nonlinear evolutionary equation, known ple, for a period of 10 min and an Alfvén speed of 1 Mm/s, the as the Cohen–Kulsrud equation (Cohen & Kulsrud 1974). The longitudinal wavelength is 600 Mm. In a plane wave, the trans- cubically-nonlinear term in this equation accounts for the non- verse wavelength should be much larger than the longitudinal linear self-interaction of linearly or elliptically polarised, plane wavelength. Hence, for the generation of a plane Alfvén wave of Alfvén waves via the wave-induced perturbation of the local a 10-min period, the wave driver should be of the size exceeding Alfvén speed. These perturbations are often referred to as the the solar diameter. Also there should be no transverse structur- nonlinearly-induced compressible motions in Alfvén waves. In ing of the plasma in the Alfvén speed, otherwise the wavefront contrast with the parallel magnetoacoustic waves (e.g., slow is distorted, and the wave becomes non-planar (e.g. Botha et al. waves in the low-β plasma of the solar corona), these perturba- 2000). Thus, the study of the initial stage of the nonlinear cas- tions exist even in the zero-β regime. Circularly polarised plane cade in the corona requires consideration of non-planar Alfvén Alfvén waves are not subject to this effect. waves. Article published by EDP Sciences A80, page 1 of 4 A&A 526, A80 (2011)

In non-uniform plasma structures, Alfvén waves are situ- force connected with the azimuthal rotation of the plasma, the ated on magnetic surfaces. In the field-aligned structured coro- magnetic tension force caused by the magnetic field curvature, nal plasmas, Alfvén waves can be present in the form of tor- and the ponderomotive force that is connected with the longitu- sional modes (see discussion in Van Doorsselaere et al. 2008a,b). dinal gradients of the magnetic pressure perturbation in the tor- Torsional waves are intensively studied theoretically in the con- sional wave. The first two forces are absent from the plane wave text of coronal heating (e.g. Antolin et al. 2008; Copil et al. 2008; theory of Alfvén waves and appear because of plasma structur- Antolin & Shibata 2010), coronal seismology (Zaqarashvili & ing. These forces can modify the flux-tube cross-sectional area, Murawski 2007; Verth et al. 2010) and particle acceleration in hence inducing compressible plasma motions in the longitudi- solar flares (Fletcher & Hudson 2008). There is some indirect nal and radial directions. In the second-order thin flux-tube ap- evidence of torsional standing modes (Zaqarashvili 2003)and proximation, these effects are taken into account in the trans- propagating waves (Banerjee et al. 2009) in spectroscopic data, verse force-balance equation. The ponderomotive force causes and also in microwave emission (Tapping 1983; Grechnev et al. the nonlinear self-interaction of Alfvén waves in the Cohen- 2003). The aim of this article is to study compressible pertur- Kulsrud equation formalism. bations induced by long-wavelength weakly-nonlinear torsional We consider a weakly-nonlinear torsional wave and restrict waves, which are essentially non-plane. our attention to the linear terms of the compressible variables, described by the equations 2. Compressible flows induced by torsional waves 1 A ρ ∂V A B ∂2B p + B B − 0 0 − 0 z0 z π z0 z π ∂ π2 ∂ 2 We are interested in torsional waves of wavelength much longer 4 2 t 16 z 2 2 than the diameter of the wave-guiding magnetic flux tube with A0 J A0 ρ0 Ω = pext + − , (4) cylindrical coordinates (r,ϕ,z). In an untwisted and non-rotating T 8π2 2π tube with the equilibrium magnetic field Bz0, mass density ρ0, and circular cross-section area A , linear torsional waves are 0 ∂ ∂ ∂ twisting azimuthal motions vϕ accompanied by the appearance u p 1 2 J ρ0 + = − JR , (5) of the azimuthal component of the magnetic field Bϕ.Atthe ∂ t ∂ z 4π ∂z axis of the flux-tube, both quantities vanish, and hence can- where pext is the total pressure in the external medium, and not be described by the first order thin magnetic flux theory √ T = /π of Roberts & Webb (1978), while they appear in the second- R A0 is the flux-tube radius. The nonlinear terms associ- order thin flux-tube approximation of Zhugzhda (1996). The ated with the torsional wave are on the right hand side. In Eq. (5) approximation allows one to describe axisymmetric (m = 0, the term responsible for the ponderomotive force appears after where m is the azimuthal wave number) flows of plasma, in- accounting for the higher-order terms in the thin flux-tube ex- cluding sausage, longitudinal and torsional modes (Zhugzhda & pansion. Nakariakov 1999; Vasheghani Farahani et al. 2010) and their in- The compressible variables are expressed through the lon- teraction. Linear torsional waves are governed by the equations gitudinal component of the magnetic field perturbation Bz, with for the quantities Ω=vϕ/r and J = Bϕ/r, which in the thin the use of the linear expressions flux tube approximation correspond to the vorticity and electric ∂ρ ∂ u current density, respectively, + ρ + 2ρ V = 0, (6) ∂ t 0 ∂ z 0 ∂ Ω ∂ − Bz0 J = , ∂ πρ ∂ 0 (1) t 4 0 z ∂ B z + 2B V = 0, (7) ∂ t z0 ∂ J ∂ Ω − B = 0, (2) ∂ t z0 ∂ z ∂ p ∂ρ − C2 = 0, (8) where r is the radial coordinate. Also it is worth mentioning that ∂ t s ∂ t in the magnetic cylinder, a torsional wave could exist on any cylindrical shell (magnetic surface), having an arbitrary depen- where Cs is the sound speed. Equations (6)–(8) are readily com- dence on r, provided vϕ and Bϕ are zero on the axis of the cylin- bined in a driven wave equation for the density perturbation, der. In the thin flux tube approximation, those dependencies are A ∂2 pext A ∂2 J2 approximated by linear functions. Equations (1)and(2) are read- (C2 + C2 )D ρ + 0 D D ρ = T + 0 − ρ Ω2 s A T 4π s A ∂t2 2π ∂t2 4π 0 ily combined in the wave equation, R2C2 ∂ ∂J A R2 ∂ ∂J ∂2 ∂2 + A J + 0 D J ,(9) − C2 J = 0, (3) 4π ∂z ∂z 16π2 A ∂z ∂z ∂t2 A ∂z2 where where CA = Bz0/ 4πρ0 is the Alfvén speed. In the linear ∂2 ∂2 2 2 regime, these motions are decoupled from compressible mo- 2 CACs D , , = − C , and C = · (10) tions. The latter are described by the variables u and V, the lon- T s A ∂ 2 T, s, A ∂ 2 T 2 2 t z C + Cs gitudinal and radial components of the velocity, respectively, ρ A the mass density, Bz the longitudinal component of the magnetic The last term on the right hand side of Eq. (9) can be ne- field, p the gas pressure and A the perturbation of the cross- glected in comparison with the other terms, as it is proportional 2 sectional area of the flux tube. to A0/λ 1, where λ is the longitudinal wavelength. The A long-wavelength torsional wave of a finite amplitude in- second term on the left hand side, responsible for wave disper- duces compressible motions by three forces: the centrifugal sion, can be neglected too. Equation (9) describes the excitation

A80, page 2 of 4 S. Vasheghani Farahani et al.: Nonlinear long-wavelength torsional Alfvén waves of compressible motions by weakly nonlinear, long-wavelength Following the formalism developed in Nakariakov et al. (2000a), torsional waves in a thin magnetic flux tube. In the following we obtain the equivalent of Eq. (11) we ignore the perturbation of the total pressure in the external medium, pext, concentrating on the compressible flows inside the 1 ∂ ∂By T D ρ = By · (17) flux-tube. s 4π ∂z ∂z A ∂2 J2 = ω − (C2 + C2 )D ρ = 0 − ρ Ω2 Taking By Byacos( t kz)weget s A T 2π ∂t2 4π 0 2 2 1 2 2 R C ∂ ∂J D ρ = − By k cos(2ωt − 2kz), (18) + A J , (11) s 4π a 4π ∂z ∂z with the solution also, the back-reaction of the induced compressible flows on the 2 2 torsional waves through the modification of the local Alfvén Byak speed and the flux-tube diameter is not considered. The latter ρ = cos(2ωt − 2kz), (19) 16π(ω2 − C2 k2β) assumption is justified by the consideration of the quadratically A nonlinear terms only, while the consideration of the Alfvén wave β = 2/ 2 ≈ / self-interaction appears when the cubic nonlinearity is taken into where Cs CA. With ja Bϕa R, we observe that the right account. hand side terms in Eqs. (16)and(19) are of the similar order. ff The first term on the right hand side of Eq. (11)hastwo However, there is an important di erence between solutions (16) terms associated with the nonlinear torsional wave, which have and (19) is that in the case of torsional waves, there is no possi- opposite signs. Hence, their combined effect on the compress- bility for a resonance of the Alfvén waves with the sound wave. β ible flows depends upon the phase relations between the twist Note, in the zero- limit, both solutions coincide. and the rotation of the plasma in the torsional waves. Consider separately the cases of propagating and standing waves. 4. Standing torsional waves We consider standing torsional waves that may appear in closed 3. Propagating torsional waves magnetic structures, e.g. coronal loops with k = π/L where L is We take a propagating solution of Eq. (11), the loop length = ω , J = ja cos(ω t − kz), (12) J 2 ja cos( t)cos(kz) ja ω Ω=−2 sin(ω t)sin(kz), (20) where ja is the amplitude of the magnetic twist, and and k (4πρ )1/2 are the frequency and the wavenumber, respectively, for which 0 ω = ±CAk.WeuseEq.(1) to express the associated rotation of where the phase relations between the perturbed magnetic twist the flux-tube as and the vorticity are obtained from Eqs. (1)and(2). Substituting ff j Eqs. (20)inEq.(11), and neglecting dispersive e ects, higher Ω=− a cos(ω t − kz). (13) order terms and the perturbations of the external medium, we πρ 1/2 (4 0) obtain For such a solution, the first term on the right-hand side of R2C2 Eq. (11)vanishes: C2 + C2 D ρ = − A j2k2 ( + ωt ) kz ( s A) T π a 1 cos(2 ) cos(2 ) 2 J2 j2 (− j )2 A j2ω2 − ρ Ω2 = a − ρ a 2 ω t − kz = . − 0 a ωt . 0 0 cos ( ) 0 (14) 2 cos(2 ) (21) 4π 4π 4πρ0 π This means that the effects of nonlinear magnetic twist and The first term on the right hand side represents the ponderomo- plasma rotation in the travelling wave cancel out each other, and tive effect, and the second term contains the magnetic tension do not add new effects to the nonlinear compressibility of propa- and centrifugal effects. gating torsional waves. Thus, the induced compressible motions In the finite-β case with the constraint β 1 the solution for are described by the equation Eq. (21)is 2 2 R2 j2 2 2 R CA 2 2 a (C + C )D ρ = − k j cos[2(ωt − kz)]. (15) ρ = − (1 − cos(2Cskt)) cos(2kz) s A T π a πC2 + β 4 ⎛8 s (1 ) ⎞ ⎜ R2 R2cos(2kz)⎟ with the driven solution + ⎝⎜ + ⎠⎟ × j2cos(2ωt), (22) 4πC2 (1 + β) 8πC2 a R2 j2 A A ρ = a ω − . cos[2( t kz)] (16) ω = 16πC2 where CAk. Equation (22) is similar to Eq. (13) of A Tikhonchuk et al. (1995) obtained for shear Alfvén waves. Thus, we obtain that the right hand side of Eq. (16) is indepen- According to Eq. (22), standing torsional waves induce dent of the value of the sound speed. growing compressible perturbations, similarly to standing shear Compare Eq. (16) with the case without transverse structur- Alfvén waves (Tikhonchuk et al. 1995; Verwichte et al. 1999; ing, i.e. with plane shear Alfvén waves. Consider waves propa- Litwin & Rosner 1998) and standing kink modes of coronal gating in the z-direction, taking ∂/∂y = 0and∂/∂x = 0. Restrict loops (Terradas & Ofman 2004). The growth is connected with our attention to the linearly polarised Alfvén waves, vy and By. the ponderomotive term on the right hand side of Eq. (21).

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Finite-β effects cause saturation of the compressible perturba- waves are similar to those derived for plane shear Alfvén waves. tion. Using the results obtained for shear Alfvén waves, we ob- For propagating waves, the efficiency of the nonlinear generation tain that the highest value of the density perturbation, of compressible perturbations does not grow with the plasma beta. This effect should be taken into account in one-dimensional ρ 2 2 R ja 1 models of the solar and stellar wind acceleration by Alfvén = , (23) ρ πρ 2 + 2/ 2 waves. 0 4 0 Cs (1 Cs CA) / is reached at the time tmax L 2Cs,whereL is the loop length. Acknowledgements. T.V.D. is a post-doctoral researcher of the FWO – Also, in the case of standing waves, the centrifugal and mag- Vlaanderen. netic tension terms do not cancel each other, and produce compressible perturbations oscillating at the double frequency of the torsional wave. However, those terms do not cause the secular References growth of compressible perturbations. Antolin, P., & Shibata, K. 2010, ApJ, 712, 494 Antolin, P., Shibata, K., Kudoh, T., Shiota, D., & Brooks, D. 2008, ApJ, 688, 669 5. Conclusions Banerjee, D., Pérez-Suárez, D., & Doyle, J. G. 2009, A&A, 501, L15 Bisnovatyi-Kogan, G. S. 2007, MNRAS, 376, 457 We considered compressible perturbations induced in straight Botha, G. J. J., Arber, T. D., Nakariakov, V. M., & Keenan, F. P. 2000, A&A, untwisted and non-rotating magnetic flux tubes by weakly- 363, 1186 nonlinear long-wavelength torsional waves. We can summarise Charbonneau, P., & MacGregor, K. B. 1995, ApJ, 454, 901 Cohen, R. H., & Kulsrud, R. M. 1974, Phys. Fluids, 17, 2215 our findings as follows: Copil, P., Voitenko, Y., & Goossens, M. 2008, A&A, 478, 921 Cranmer, S. R. 2009, Living Rev. Sol. Phys., 6, 3 1. Long-wavelength torsional waves induce compressible per- Fletcher, L., & Hudson, H. S. 2008, ApJ, 675, 1645 turbations by the ponderomotive, centrifugal and magnetic Grechnev, V. V., White, S. M., & Kundu, M. R. 2003, ApJ, 588, 1163 twist forces. The perturbations have double the frequency of Litwin, C., & Rosner, R. 1998, ApJ, 506, L143 the inducing torsional wave. The efficiency of the excitation Nakariakov, V. M., Mendoza-Briceño, C. A., & Ibáñez S., M. H. 2000a, ApJ, 528, 767 depends upon the spatial (standing and propagating) struc- Nakariakov, V. M., Ofman, L., & Arber, T. D. 2000b, A&A, 353, 741 ture of the inducing torsional wave. Ofman, L. 2005, Space Sci. Rev., 120, 67 2. The efficiency of the generation of compressible perturba- Ofman, L., & Davila, J. M. 1998, J. Geophys. Res., 103, 23677 tions by long-wavelength torsional waves is independent of Roberts, B., & Webb, A. R. 1978, Sol. Phys., 56, 5 the plasma-β (see Eq. (16)). This result is different from Suzuki, T. K. 2004, MNRAS, 349, 1227 Suzuki, T. K. 2007, ApJ, 659, 1592 the excitation of compressible perturbations by plane shear Suzuki, T. K. 2008, Nonlinear Proc. Geophys., 15, 295 Alfvén waves, in which case the efficiency grows when the Suzuki, T. K., & Inutsuka, S. 2005, ApJ, 632, L49 Alfvén and sound speeds approach each other. The discrep- Suzuki, T. K., Sumiyoshi, K., & Yamada, S. 2008, ApJ, 678, 1200 ancy is connected with the fact that the tube speed is always Tapping, K. F. 1983, Sol. Phys., 87, 177 Terradas, J., & Ofman, L. 2004, ApJ, 610, 523 lower than the Alfvén speed. The relative amplitude of the Tikhonchuk, V. T., Rankin, R., Frycz, P., & Samson, J. C. 1995, Phys. Plasmas, ρ/ρ = 2/ 2 induced density perturbation is 0 Bϕ 4Bz0,whereBϕ is 2, 501 the perturbation of the magnetic field at the boundary of the Torkelsson, U., & Boynton, G. C. 1998, MNRAS, 295, 55 flux tube. Van Doorsselaere, T., Brady, C. S., Verwichte, E., & Nakariakov, V. M. 2008a, 3. There are two kinds of compressible perturbations induced A&A, 491, L9 Van Doorsselaere, T., Nakariakov, V. M., & Verwichte, E. 2008b, ApJ, 676, by standing torsional waves: the perturbations which grow L73 with the time scale 1/2Csk,wherek is the longitudinal wave Vasheghani Farahani, S., Nakariakov, V. M., & Van Doorsselaere, T. 2010, A&A, number of the torsional wave, and the perturbations oscillat- 517, A29 ing at the double frequency of the driving torsional mode. Verth, G., Erdélyi, R., & Goossens, M. 2010, ApJ, 714, 1637 Verwichte, E., Nakariakov, V. M., & Longbottom, A. W. 1999, J. Plasma Phys., The growing density perturbations saturate at the level in- 62, 219 versely proportional to the sound speed. Zaqarashvili, T. V. 2003, A&A, 399, L15 Zaqarashvili, T. V., & Murawski, K. 2007, A&A, 470, 353 Thus we conclude that nonlinear compressible effects which ac- Zhugzhda, Y. D. 1996, Phys. Plasmas, 3, 10 company standing weakly-nonlinear long-wavelength torsional Zhugzhda, Y. D., & Nakariakov, V. M. 1999, Phys. Lett. A, 252, 222

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